# Constructing algebras from a vector space

When applied to two copies of the same vector space $${V}$$, the tensor product is sometimes called the “outer product,” since it is a linear map from two vectors to a “bigger” object “outside” $${V}$$, as opposed to the inner product, which is a linear map from two vectors to a “smaller” object “inside” $${V}$$.

The term “outer product” also sometimes refers to the exterior product $${V\wedge V}$$ (AKA wedge product, Grassmann product), which is defined to be the tensor product $${V\otimes V}$$ modulo the relation $${v\wedge v\equiv0}$$. By considering the quantity $${(v_{1}+v_{2})\wedge(v_{1}+v_{2})}$$, this immediately leads to the equivalent requirement of identifying anti-symmetric elements, $${v_{1}\wedge v_{2}\equiv-v_{2}\wedge v_{1}}$$.

These outer products can be applied to a vector space $${V}$$ to generate a larger vector space, which is then an algebra under the outer product. If $${V}$$ has a pseudo inner product defined on it, we can then generalize it to the larger algebra.